Independence complexes of $(n \times 4)$ and $(n \times 5)$-grid graphs

TL;DR

This paper determines the homotopy types of independence complexes for specific grid graphs, showing they are homotopy equivalent to wedges of spheres, advancing understanding in topological combinatorics.

Contribution

It provides explicit homotopy type characterizations for independence complexes of (n×4) and (n×5) grid graphs, a novel result in topological graph theory.

Findings

Independence complexes are homotopy equivalent to wedges of spheres.

Homotopy types are explicitly determined for (n×4) and (n×5) grid graphs.

Results contribute to the understanding of topological properties of grid graph complexes.

Abstract

We determine the homotopy types of the independence complexes of $(n \times 4)$ and $(n \times 5)$-square grid graphs. In fact, they are homotopy equivalent to wedges of spheres.